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{\bf
In how many ways can you Rectangle a Rectangle?
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{\it Pablo BLANCO, Robert DOUGHERTY-BLISS, Natalya TER-SAAKOV, and Doron Zeilberger}

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{\bf Abstract}: There are $2$ to the power $n-1$ ways to tile a $1 \times n$ rectangle with rectangular tiles (of any length, of course they all must have width $1$), but
in how many ways can you tile  a $100 \times 1000$  rectangle with rectangular tiles?. Neither humankind, nor computer-kind, will (most probably) ever know the exact number.
But we can compute these numbers for $m \times n$ rectangles if $m$ is not too big, while $n$ can be as big as one wishes.
We also consider, {\it weighted counting}, by the number of participating tiles, and the number of edges (of the grid) that participate in the tiling.



{\bf Preface: How it all started} 



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{\bf References}


[Sl] Neil Sloane, The On-Line encyclopedia of integer sequences, {\tt https://oeis.org/} . Sequences

[St] Richard P. Stanley, {\it Enumerative Combinatorics}, Volume 1, Wadsworth \& Brooks/Cole, (first edition), 1986.



[W] Herbert S. Wilf, A unified setting for sequencing, ranking, and selection algorithms for combinatorial objects, Advances in Math 24 (1977) , 281-291. \hfill\break
{\tt https://www2.math.upenn.edu/\~{}wilf/website/Unified\%20setting.pdf} \quad .
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Pablo Blanco, Natalya Ter-Saakov and Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Emails: {\tt  pablancoh at aol dot com} , {\tt  nt399 at rutgers dot edu} ,{\tt DoronZeil at gmail  dot com}   \quad .
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Robert Dougherty-Bliss, Department of Mathematics, Dartmouth College, {\tt robert dot w dot bliss at gmail dot com} 
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{\bf May 5, 2026} 

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